Output–input ratio analysis and DEA frontier

Data envelopment analysis (DEA) is a mathematical programming technique for identifying efficient frontiers for peer decision making units (DMUs). The ability of identifying frontier DMUs prior to the DEA calculation is of extreme importance to an effective and efficient DEA computation. In this paper, we present mathematical properties which characterize the inherent relationships between DEA frontier DMUs and output–input ratios. It is shown that top-ranked performance by ratio analysis is a DEA frontier point. This in turn allows identification of membership of frontier DMUs without solving a DEA program. Such finding is useful in streamlining the solution of DEA.     

Profit maximization problem for Cobb–Douglas and CES production functions

Production functions are used to model the production activity of enterprises. In this article, we formulate the necessary and sufficient conditions of strict concavity for Cobb–Douglas and constant elasticity of substitution (CES) production functions. These conditions constitute the theoretical foundation for analyzing the profit maximization problem. An optimal solution is constructed in analytical form and some of its properties are described. Three approaches to solving the profit maximization problem are considered and their equivalence is established. For a Cobb–Douglas production function we investigate the dependence of the maximum profit on elasticity coefficients. A similar analysis is carried out also for the CES production function. The article presents a systematic and detailed discussion of the relevant topics. The topic is related to the investigation of innovation activity of enterprises. The theoretical results and the explicit analytical relationships provide a theoretical and algorithmic base for the “Planer” optimization software—a useful product for the analysis of the production activity of enterprises modeled by production function tools.     

A geometric perspective on the generalized Cobb–Douglas production functions

In this work we obtain an interesting link between some fundamental notions in the theory of production functions and the differential geometry of hypersurfaces in Euclidean spaces. In fact we establish that a generalized Cobb–Douglas production function has decreasing/increasing return to scale if and only if the corresponding hypersurface has positive/negative Gaussian curvature. Moreover, this production function has constant return to scale if and only if the corresponding hypersurface is developable.     

An alternative approach to solving cost minimization problem with Cobb–Douglas technology

Central European Journal of Operations Research - We propose a new method for solving the production cost minimization problem with Cobb–Douglas technology. The method is based on weighted...     

Cobb—Douglas生产函数参数估计的加权方法

本文给出了Cobb-Douglas生产函数参数估计的加权方法,实例分析表明,本文方法所得模型的精度明显高于经典方法所得模型的精度。     

An optimal investment,consumption, leisure,and voluntary retirement problem with Cobb–Douglas utility: Dynamic programming approaches

We consider an optimal consumption, leisure, investment, and voluntary retirement problem for an agent with a Cobb–Douglas utility function. Using dynamic programming, we derive closed form solutions for the value function and optimal strategies for consumption, leisure, investment, and retirement.     

Geometric Arveson–Douglas conjecture

We prove the Arveson–Douglas essential normality conjecture for graded Hilbert submodules that consist of functions vanishing on a given homogeneous subvariety of the ball, smooth away from the origin. Our main tool is the theory of generalized Toeplitz operators of Boutet de Monvel and Guillemin.     

On the Douglas–Dirichlet functional and harmonic quasiconformal mappings

In this article, we discuss the minimal mappings of Douglas–Dirichlet functional and harmonic quasiconformal mappings, and solve the uniqueness problem of harmonic quasiconformal mappings posed by Shibata.     

Optimal taxation of a perfectly competitive firm with Cobb–Douglas production function as a bilevel programming problem

Central European Journal of Operations Research - Taxation is one of the most powerful instruments of fiscal policy, affecting economic growth and investments, as well as competitiveness of...     

On the Douglas–Rachford algorithm

Mathematical Programming - The Douglas–Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. The behaviour of the...     

Geometric Arveson–Douglas Conjecture-Decomposition of Varieties

In this paper, we prove the Geometric Arveson–Douglas Conjecture for a special case that allows some singularity on \(\partial {\mathbb {B}_n}\). More precisely, we show that if a variety can be decomposed into two varieties, each having nice properties and intersecting nicely with \(\partial \mathbb {B}_n\), then the Geometric Arveson–Douglas Conjecture holds on this variety. We obtain this result by applying a result by Suárez, which allows us to “localize” the problem. Our result then follows from the simple case when the two varieties are intersection of linear subspaces with \(\mathbb {B}_n\).     

A Cyclic Douglas–Rachford Iteration Scheme

In this paper, we present two Douglas–Rachford inspired iteration schemes which can be applied directly to N-set convex feasibility problems in Hilbert space. Our main results are weak convergence of the methods to a point whose nearest point projections onto each of the N sets coincide. For affine subspaces, convergence is in norm. Initial results from numerical experiments, comparing our methods to the classical (product-space) Douglas–Rachford scheme, are promising.     

Local Convergence Properties of Douglas–Rachford and Alternating Direction Method of Multipliers

The Douglas–Rachford and alternating direction method of multipliers are two proximal splitting algorithms designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local linear convergence behaviour of Douglas–Rachford (resp. alternating direction method of multipliers) when the involved functions (resp. their Legendre–Fenchel conjugates) are moreover partly smooth. More precisely, when the two functions (resp. their conjugates) are partly smooth relative to their respective smooth submanifolds, we show that Douglas–Rachford (resp. alternating direction method of multipliers) (i) identifies these manifolds in finite time; (ii) enters a local linear convergence regime. When both functions are locally polyhedral, we show that the optimal convergence radius is given in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified submanifolds. Under polyhedrality of both functions, we also provide conditions sufficient for finite convergence. The obtained results are illustrated by several concrete examples and supported by numerical experiments.     

Global convergence of a non-convex Douglas–Rachford iteration

We establish a region of convergence for the proto-typical non-convex Douglas–Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration Borwein and Sims (Fixed-point algorithms for inverse problems in science and engineering, pp. 93–109, 2011) was only able to establish local convergence, and was ineffective in that no explicit region of convergence could be given.     

Comparing frontier methods for economic–environmental trade-off analysis

This paper uses a mechanistic frontier approach as a reference to evaluate the ability of conventional parametric (SFA) and non-parametric (DEA) frontier approaches for analyzing economic–environmental trade-offs. Conventional frontier approaches are environmentally adjusted through incorporating the materials balance principle. The analysis is worked out for the Flemish pig finishing case, which is both representative and didactic. Results show that, on average, SFA and DEA yield adequate economic–environmental trade-offs. Both methods are good estimators for technical efficiency. Cost allocative and environmental allocative efficiency scores are less robust, due to the well-known methodological advantages and disadvantages of SFA and DEA. For particular firms, SFA, DEA and the mechanistic approach may yield different economic–environmental trade-offs. One has therefore to be careful when using conventional frontier approaches for firm-specific decision support. The mechanistic approach allows for optimizing performances per average present finisher, which is the production unit in pig finishing. Conventional frontier methods do not allow for this optimization since the number of average present finishers varies along the production functions. Since the mechanistic production function is based on underlying growth, feed uptake and mortality functions, additional firm-specific indicators can also be calculated at each point of the production function.     

A generalization of Cachazo–Douglas–Seiberg–Witten conjecture for symmetric spaces

We extend the original Cachazo–Douglas–Seiberg–Witten conjecture on the structure of the chiral ring of classical supersymmetric Yang-Mills theory to symmetric spaces.     

A unified Douglas–Rachford algorithm for generalized DC programming

Journal of Global Optimization - We consider a class of generalized DC (difference-of-convex functions) programming, which refers to the problem of minimizing the sum of two convex (possibly...